In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may...

called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists...

linear algebra concerns the description of K-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in K-Vect correspond...

the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure...

affine space with a distinguished point O may be identified with its associated vector space (see Affine space § Vector spaces as affine spaces), the preceding...

complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano...

three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are...

defining a projective space as the set of the vector lines in a vector space of dimension one more. As for affine spaces, projective spaces are defined over...

metric spaces need not be the same) And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are...

high-dimensional Euclidean space. (Use the Whitney embedding theorem.) Take a small neighborhood of M in that Euclidean space, Nε. Extend the vector field...

Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole...

basis Cyclic decomposition theorem Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping or Galilean...

In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called...

nullity of f (the dimension of the kernel of f). It follows that for linear transformations of vector spaces of equal finite dimension, either injectivity...

very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable...

primarily in three-dimensional Euclidean space, R 3 . {\displaystyle \mathbb {R} ^{3}.} The term vector calculus is sometimes used as a synonym for the broader...

are one-dimensional spaces but are usually referred to by more specific terms. Any field K {\displaystyle K} is a one-dimensional vector space over itself...

finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe...

geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions...

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through...

mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes...

vector space V have the same cardinality, which is called the dimension of V; this is the dimension theorem for vector spaces. Moreover, two vector spaces...

isomorphism theorem for affine spaces. Affine spaces are usually studied by analytic geometry using coordinates, or equivalently vector spaces. They can...

straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general...

Erdős–Kaplansky theorem is a theorem from functional analysis. The theorem makes a fundamental statement about the dimension of the dual spaces of infinite-dimensional...

Cayley–Hamilton theorem (Linear algebra) Dimension theorem for vector spaces (vector spaces, linear algebra) Euler's rotation theorem (geometry) Exchange theorem (linear...

with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications...

spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology...

{\displaystyle {\overline {W}}.} The vector spaces V {\displaystyle V} and V ¯ {\displaystyle {\overline {V}}} have the same dimension over the complex numbers and...

coordinate between the points. The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects...